On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints,Part 1: Theory

In a companion paper (Cromvik and Patriksson, Part I, J. Optim. Theory Appl., 2010), the mathematical modeling framework SMPEC was studied; in particular, global optima and stationary solutions to SMPECs were shown to be robust with respect to the underlying probability distribution under certain assumptions. Further, the framework and theory were elaborated to cover extensions of the upper-level objective: minimization of the conditional value-at-risk (CVaR) and treatment of the multiobjective case. In this paper, we consider two applications of these results: a classic traffic network design problem, where travel costs are uncertain, and the optimization of a treatment plan in intensity modulated radiation therapy, where the machine parameters and the position of the organs are uncertain. Owing to the generality of SMPEC, we can model these two very different applications within the same framework. Our findings illustrate the large potential in utilizing the SMPEC formalism for modeling and analysis purposes; in particular, information from scenarios in the lower-level problem may provide very useful additional insights into a particular application.     

A regularized decomposition method for minimizing a sum of polyhedral functions

A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.On leave from Instytut Automatyki, Politechnika Warszawska, Poland.     

On stability in multiobjective programming — A stochastic approach

In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behaviour of the constraint sets, the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints, where (i) the distribution function of the random variable is estimated by the empirical distribution function, (ii) certain parameters have to be estimated.     

Solution approaches for the multiobjective stochastic programming

We survey in this paper various solution approaches for multiobjective stochastic problems where random variables can be in both objectives and constraints parameters. Once a problem requires a stochastic formulation, a first step consists in transforming the problem into its deterministic formulation. We propose to classify and evaluate such transformations with regards to the many proposed concepts of efficiency. The paper addresses also some applications of the multiobjective stochastic programming models.     

Concavity cuts for disjoint bilinear programming

We pursue the study of concavity cuts for the disjoint bilinear programming problem. This optimization problem has two equivalent symmetric linear maxmin reformulations, leading to two sets of concavity cuts. We first examine the depth of these cuts by considering the assumptions on the boundedness of the feasible regions of both maxmin and bilinear formulations. We next propose a branch and bound algorithm which make use of concavity cuts. We also present a procedure that eliminates degenerate solutions. Extensive computational experiences are reported. Sparse problems with up to 500 variables in each disjoint sets and 100 constraints, and dense problems with up to 60 variables again in each sets and 60 constraints are solved in reasonable computing times. Received: October 1999 / Accepted: January 2001?Published online March 22, 2001     

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.     

An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

概率约束最优化问题

概率约束最优化问题是随机规划的一类重要问题,在金融、管理和工程计划等领域有广泛的应用. 概率约束优化问题近年来受到了广泛的关注和重视,在应用建模、理论和方法等方面取得了不少重要的进展. 这里主要概述和总结处理概率约束的主要方法和思想,包括凸内逼近方法、情景逼近方法、DC方法和整数规划方法等,并对概率约束最优化的研究前景进行讨论.     

Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.     

Mobility allowance shuttle transit (MAST) services: MIP formulation and strengthening with logic constraints

We study a hybrid transportation system referred to as mobility allowance shuttle transit (MAST) where vehicles may deviate from a fixed path consisting of a few mandatory checkpoints to serve demand distributed within a proper service area. In this paper we propose a mixed integer programming (MIP) formulation for the static scheduling problem of a MAST type system. Since the problem is NP-Hard, we develop sets of logic cuts, by using reasonable assumptions on passengers’ behavior. The purpose of these constraints is to speed up the search for optimality by removing inefficient solutions from the original feasible region. Experiments show the effectiveness of the developed inequalities, achieving a reduction up to 90% of the CPU solving time for some of the instances.     

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.     

Primal—Dual Constraint Aggregation with Application to Stochastic Programming

The special constraint structure and large dimension are characteristic for multistage stochastic optimization. This results from modeling future uncertainty via branching process or scenario tree. Most efficient algorithms for solving this type of problems use certain decomposition schemes, and often only a part of the whole set of scenarios is taken into account in order to make the problem tractable.We propose a primal–dual method based on constraint aggregation, which constructs a sequence of iterates converging to a solution of the initial problem. At each iteration, however, only a reduced sub-problem with smaller number of aggregate constraints has to be solved. Number of aggregates and their composition are determined by the user, and the procedure for calculating aggregates can be parallelized. The method provides a posteriori estimates of the quality of the current solution approximation in terms of the objective function value and the residual.Results of numerical tests for a portfolio allocation problem with quadratic utility function are presented.     

Global Nonparametric Sufficient Optimality Conditions for Semi-Infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η, ρ)-Invex Functions

In this paper, we establish a set of necessary optimality conditions and discuss a fairly large number of sets of global nonparametric sufficient optimality criteria under various generalized (η, ρ)-invexity assumptions for a minmax fractional programming problem with infinitely many nonlinear inequality and equality constraints.     

Optimal Security Liquidation Algorithms

This paper develops trading strategies for liquidation of a financial security, which maximize the expected return. The problem is formulated as a stochastic programming problem that utilizes the scenario representation of possible returns. Two cases are considered, a case with no constraint on risk and a case when the risk of losses associated with trading strategy is constrained by Conditional Value-at-Risk (CVaR) measure. In the first case, two algorithms are proposed; one is based on linear programming techniques, and the other uses dynamic programming to solve the formulated stochastic program. The third proposed algorithm is obtained by adding the risk constraints to the linear program. The algorithms provide path-dependent strategies, i.e., the fraction of security sold depends upon price sample-path of the security up to the current moment. The performance of the considered approaches is tested using a set of historical sample-paths of prices.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

Minimizing the sum of a linear and a linear fractional function applying conic quadratic representation: continuous and discrete problems

This paper tries to minimize the sum of a linear and a linear fractional function over a closed convex set defined by some linear and conic quadratic constraints. At first, we represent some necessary and sufficient conditions for the pseudoconvexity of the problem. For each of the conditions, under some reasonable assumptions, an appropriate second-order cone programming (SOCP) reformulation of the problem is stated and a new applicable solution procedure is proposed. Efficiency of the proposed reformulations is demonstrated by numerical experiments. Secondly, we limit our attention to binary variables and derive a sufficient condition for SOCP representability. Using the experimental results on random instances, we show that the proposed conic reformulation is more efficient in comparison with the well-known linearization technique and it produces more eligible cuts for the branch and bound algorithm.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

Solving nonlinear principal-agent problems using bilevel programming

While significant progress has been made, analytic research on principal-agent problems that seek closed-form solutions faces limitations due to tractability issues that arise because of the mathematical complexity of the problem. The principal must maximize expected utility subject to the agent’s participation and incentive compatibility constraints. Linearity of performance measures is often assumed and the Linear, Exponential, Normal (LEN) model is often used to deal with this complexity. These assumptions may be too restrictive for researchers to explore the variety of relationships between compensation contracts offered by the principal and the effort of the agent. In this paper we show how to numerically solve principal-agent problems with nonlinear contracts. In our procedure, we deal directly with the agent’s incentive compatibility constraint. We illustrate our solution procedure with numerical examples and use optimization methods to make the problem tractable without using the simplifying assumptions of a LEN model. We also show that using linear contracts to approximate nonlinear contracts leads to solutions that are far from the optimal solutions obtained using nonlinear contracts. A principal-agent problem is a special instance of a bilevel nonlinear programming problem. We show how to solve principal-agent problems by solving bilevel programming problems using the ellipsoid algorithm. The approach we present can give researchers new insights into the relationships between nonlinear compensation schemes and employee effort.     

求解单阶段随机规划的一种光滑逼近法

本文借助某种离散方式把单阶段随机规划问题转化为具有多个约束的确定性非线性规划，然后利用极大熵函数方法，把此确定性规划转化为只带简单约束的非线性规划，由此提出了求解这种随机规划的光滑逼近法，同时给出了该法的收敛性分析，较好地克服了因提高离散精度导致约束函数个数迅速增大所带来的求解困难．